[seqfan] Re: [math-fun] Triangular+Triangular = Factorial

[Georgi Guninski]

On Fri, Sep 10, 2010 at 01:29:15PM -0600, Richard Guy wrote:
> Just cleaning up old email, so this is a very,
> very belated response, which may have been taken
> care of by someone long ago. Copied to seqfans,
> in case someone wants to make a sequence or
> two out of it. Solutions of
> 
>            x(x+1)/2 + y(y+1)/2 = z!
> 
> are solutions of  (2x+1)^2 + (2y+1)^2 = 8(z!) + 2
> and the first few values of  z for which there
> are solutions can easily be ascertained:
> 
> (x,y,z) = (0,1,0), (0,1,1), (1,1,2), (0,3,3), (2,2,3),
> (2,6,4), (0,15,5), (5,14,5), (45,89,7), (89,269,8),
> (210,825,9), (760,2610,10), (1770,2030,10),
> none for  z = 11, 12, one for  z = 13
> (see Ed's solution below), none for  z = 14,
> two for  z = 15 (see below), one for  z = 16,
> two for  z = 17, none for  z = 18, 19, 20,
> two for  z = 21, none for  z = 22, 23,
> two for  z = 24, none for  z = 25, 26,
> eight for  z = 27, one for  z = 28,
> two for  z = 29, none for  z = 30, 31,
> four for  z = 32, 33, sixteen for  z = 34,
> none for  z = 35, 36, ..., 41,
> two for  z = 42, none for  z = 43, 44, ..., 48,
> sixteen for  z = 49, none for  z = 50, 51, 52, 53,
> one for  z = 54, none for  z = 55, 56, ..., 65,
> two for z = 66, none for  z = 67,
> sixteen for  z = 68, ...
> (E&OE, and PARI is slowing down a bit now: AND it would
> take rather longer to find the actual solutions!)   R.
>

based on partial factorization and condition for sum of two squares i get no solution for these up to 500:

2,6,11,12,14,18,19,20,22,23,25,26,30,36,37,39,43,44,45,46,47,51,52,57,58,62,64,67,75,82,84,88,89,90,97,98,101,105,106,112,113,115,116,117,121,123,124,127,132,136,137,138,142,144,146,148,150,151,153,155,156,157,158,159,161,163,167,170,172,174,175,182,185,186,188,195,196,207,210,211,215,225,226,227,228,231,232,236,237,249,251,256,257,258,261,262,263,265,266,269,270,274,278,279,282,284,285,287,288,289,292,293,295,297,298,300,301,307,311,319,320,324,331,333,334,338,339,343,346,354,355,356,359,360,365,367,368,369,372,380,381,384,387,388,390,394,395,398,399,400,401,404,405,406,414,420,421,423,424,425,427,432,433,436,437,438,439,443,445,447,449,452,454,459,462,469,470,472,474,479,486,487,488,489

(not being in the list does not guarantee existence of solution).

out of pure luck the partial factorization gives efficient solution for
these z < 500:

72 85 86 129 190 351 457 466

8*(466!)+2 turns out to be semiprime with the trivial factor 2 and a solution is:

z= 466
x,y
85222921729141617187959575405060166696609752901590573098834126416514077773565672483050911785703481006671100211933619600944851577114592678313237046865818945349302838114710435539697718222141447917430930578114622220074407508695211610306371736164795450064668439870093761076752711118677293888593919421656829973007810860061328900334076240827532287341246410697618146804845847277882926395716945739233194725114097314196296757444551218090401643910714555666090380019712201561301428818305060658173857440835805216834364891579346048405 
353364347635635922492353798850995902174183259820036577840550987269097360108122690749643828913095579748050980110024431624792584197800019580744013559701700641539111825335537380108199080847101206063607886550421219077139702125536014223259081972386862363247352145253846520002860015830997289288258259829053250633748624491432762290810540909783232164729692959961188158546084377189471443569735598862854118278547649351215732207691458976892401035418644016219740842540242500408896545424163979811327322276724910662055025936678014461329  
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